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%\author{王立庆（2019级数学与应用数学1班）}
\author{学号 \underline{\hspace{4cm}} 姓名  \underline{\hspace{4cm}} }
%\title{高等代数第六章：向量空间}
\title{第八章欧氏空间（8.1-8.2）考试 }
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\date{2023年5月10日}

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\begin{document}

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%\begin{abstract}
%%主要内容：
%7.3. 
%7.4. 
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\begin{enumerate}

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\item %1
设 $V$ 是欧氏空间，设 $\alpha,\beta\in V$. 证明 
$\lvert \langle \alpha,\beta \rangle \rvert \le \lvert \alpha \rvert \cdot \lvert \beta \rvert$. 

\vspace{0.2cm}

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\item %2
考虑欧氏空间 $V=\mathbb{R}^3$ 中的向量组 
$%\begin{eqnarray*}
\Phi = \{ 
\alpha_1 = (2,0,1), \, 
\alpha_2 = (0,2,5), \,
\alpha_3 = (2,3,0)
\}.
$%\end{eqnarray*}

\begin{enumerate}
\item  证明向量组 $\Phi$ 是 $V$ 的一个基。
\item  用斯密特正交化方法，从 $\Phi$ 出发得到一个规范正交基。
\end{enumerate}

\vspace{0.2cm}

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\item %3
设 $V$ 是欧氏空间，设 $W$ 是 $V$ 的有限维子空间。设向量 $\alpha\in V$. 
\begin{enumerate}
\item  写出向量 $\alpha$ 在 $W$ 中的正射影的定义。 
\item  在 $\mathbb{R}^3$ 中，求向量 $\alpha=(1,2,3)$ 在由向量组 $\{(1,0,0), (0,1,0))\}$ 线性张成的子空间的正射影。
\item  在 $\mathbb{R}^4$ 中，求向量 $\alpha=(1,2,3,4)$ 在由向量组 $\{(1,1,1,0), (0,1,1,1))\}$ 线性张成的子空间的正射影。
\end{enumerate}

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\end{enumerate}


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\end{document}





